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When the Coriolis parameter is low, the fluid will be mainly laminar, and be governed by the physics of the Hadley cell. However, the entire globe cannot exist as a Hadley cell; proof of this can be sought within the Law of Conservation of Angular Momentum. Given that the radius of a parcel's orbit would be zero at the poles, utilizing solely this law would yield infinite winds aloft at those locations. Given that this is quite impossible, we are forced to rely on eddy heat transport, which "takes care of angular momentum" in other ways by breaking down polar heat transport into more localized vortices. This takes effect at latitudes in the 30-90º regime, where the Coriolis parameter is too great to maintain the laminar nature of the fluid, which results in turbulence. Elsewhere towards the equator, the fluid is largely laminar, leading to Hadley cells becoming the main means of heat transport between 30ºN and 30ºS.
//This is where somebody better than me does their equation magic...
Analogues in the Tank...
Several key differences exist between the earth and our tank experiment; after all, one is a three dimensional rotating sphere comprised of solid, liquid, and gas, complete with pressure gradients amidst the fluid as well as several means of thermal heat fluxes. Meanwhile, the tank is a two-dimensional representation, and the incompressible nature of the water leads to further behavioral differences.
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Figure 2.2: The Hadley Cell in cross-section, highlighting both the Wind Patterns that result from it's formation.
In the mid latitudes, poleward motion leads to cyclonic motion under the Coriolis effect. At the boundary between the warmer mid latitude air and the cold polar air, we also observe the jet stream, the increase of wind with height as predicted by the thermal wind Equation 1.
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$\overline{v'T'} = \overline{vT\; } - \overline{v}\overline{T\;}$
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where the overbar indicate a monthly time average,
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We will use a rotating tank as an an analog to this setup, with low rotation rate mimicking the processes at play in the tropics with low coriolis force, and high rotation rate mimicking the high latitudes. Ice at the center provides the heating imbalance between 'higher' latitudes and 'lower' latitudes. In theory, at high rotation rates, flow will break down and become turbulent. The setup is shown below in Figure 2.5.
Figure 2.5: The tank setup, showing the tank and ice in cross section across the diameter, with rotation rate
which is varied.Mathinline body \Omega
In theory, the flow vs height relation for an incompressible fluid with coefficient of thermal expansion
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$2\Omega \frac{du}{dz} = \alpha g \frac{\partial T}{\partial r}$, |
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Using the above plots, we can confirm the thermal wind relationship using the Margulis equation; though solving as displayed in figure 1.91 yields a somewhat unbalanced equation, in the absence of thermistor errors, etc., the fact that the results share the same order of magnitude is enough to lend confidence that the thermal wind balance applies.
Figure 1.91:
Hadley Cells in the Atmosphere
Figure 1.92:
(Please do note that the Transient Heat Flux in Petawatts should not feature a 10^-21 factor; that should be omitted).
Using NCDC's and NCEP's data, the Transient Energy Flux across different levels of the atmosphere in January can be plotted with relative ease. As previously mentioned, the Coriolis parameter of the earth only supports a regime of Hadley cell occurrence between 0 and 30º on either side of the equator. As such, the remainder of the globe is dominated by eddy heat transport, which will be detailed later. However, in plotting the heat flux across different levels of the atmosphere (where positive flux describes heat carried northwards), one may confirm the existence of Hadley cells is this region. This is shown in figure 1.92.
Because January features winter in the northern hemisphere, a stronger temperature gradient leads to a more significant poleward heat transport than in the southern hemisphere. This is evident in our findings, given that the peak at northern latitudes in flux is far greater than in southern latitudes. The Hadley cell theory is also supported by the weak positive flux at the surface (1000 millibars) between 0º and 30º north. This is because there is a weak easterly with a slightly northerly component as air returns radially towards the equator; theoretically, there could be negative flux, but some "mixing" of northerly winds likely aids in some weak northerly flow near the surface. The majority of the heat transport occurs aloft, but the heat flux which measures the amount of heat transport at various levels, should be directly proportional to height (for obvious reasons described previously under the theory of Hadley cells) and pressure (as denser air can carry more heat per unit area). Therefore, the location of the "maximum" should be the "happy medium" between height and pressure, which we would estimate to be located around 700-850 millibars, as revealed by the graph. This is coincident with the jet stream, which is essentially the a narrow band of intense westerlies. In addition, the rising air characteristic of the Intertropical Convergence Zone at the equator is tempered by a weak high pressure dome aloft, which can be seen in the extremely weak flux towards the equator at the uppermost levels of the atmosphere. In addition, it was previously mentioned that friction with the surface would reduce the flux close to the ground, which is supported by the lower level of flux at the 1000 millibar level.
// Magical Theory and Stuff
//Verification of Thermal Wind Equation
Eddy Heat Transport in the Tank
Likewise, four thermistors were placed in the tank when the rotational speed was increased and thus the Coriolis parameter augmented to a point when the regime would shift to one of eddy heat transport. Figure 2.0 illustrates the positions of said thermistors:
Figure 2.0:
It would be anticipated that the greatest temperatures would be found at thermistor 1, and the coolest in turn at thermistor 3 or 4. Figure 2.1 depicts the trace of thermistors' reports during the duration of the experiment:
Figure 2.1:
Though difficult to discern due to the technical limitations of Wiki, it becomes clear that the warmest temperatures were, in fact, recorded at thermistor 1, with the chilliest readings located at thermistor 3. This is exactly what would be expected. Unlike the gentler slope within the laminar fluid, however, the eddy transport nature of this second trial of the experiment lead to an oscillating, varying "stair step" pattern in temperature measurements. In addition, an effort was made to determine the velocity of different points using the particle tracking software; the positions of the points sampled is shown in figure 2.2:
Figure 2.2:
The most dramatic eddy sampled using the particle-tracking software, visible in part on bottom left-hand side of figure 2.2, is quite telling apropos to the "heat flux" within the tank. Despite rotation throughout the eddies towards and away from the center, the thermal gradient within the tank suggests that heat should have a net movement towards the center. A side view of the same graph focused on the most intense eddy illustrates the "winds" embedded within this rotating swirl of fluid; positive velocities are towards the center, with negative velocities away from the center. At first a chaotic scatter of points demarcating velocities at given points (with the x and y axes marking position, with the z axis reserved for velocity), a blue parallelogram has been drawn into Figure 2.3, along with a yellow line illustrating the zero level of velocity from the graph's perspective. One can clearly note that the area of the parallelogram containing data points above the zero line is far greater than below the line, indicating a net movement towards, and thus a heat flux in the direction of, the center. This is exactly what is expected in the atmosphere, and what our group observed in the tank.
Figure 2.3:
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Margules equation.
Hadley Cells in the Atmosphere
After having developed the idea of an overturning convective cell at low latitudes to transport heat away from the equator and analyzed tank data to develop intuition for the pattern of motion, we will now look at what physical evidence we have for its existence in the atmosphere. The Hadley cell transports heat from the equator to the poles, so we will be looking at wind velocities to see where the net movement is. We know that there is a net north-to-south heat flux, so a good first guess is that this happens via meridional winds i.e. north-south winds. Looking at a plot of meridional wind speed vs. latitude for January, which corresponds to winter in the northern hemisphere, we can see that there are poleward winds at high altitudes and equatorward winds at low altitudes. The plot is shown below with warmer colors meaning a northward velocity direction and cooler colors meaning a southward velocity direction.
Air is being pushed to the pole up high and returning to the equator at the surface, so our initial guess needs to be modified to include how air travels between high altitudes to low altitudes. As seen in our last project, convection is the process that the atmosphere uses to transport heat from the surface to higher altitudes where the heat is then radiated off into space. It makes sense then that a convective plume would exist at the equator, so we must look at a plot of vertical velocity to verify that winds bring air back to the surface at higher latitudes.
The above graph shows vertical wind velocity. The cooler colors are rising columns, and the warmer colors are sinking columns. While there are adjacent rising and sinking columns spanning the whole surface of the Earth, the strongest winds occur between the equator and 30N. The strongest winds appear in the northern hemisphere because that is where it is winter in January. When it is winter in the southern hemisphere, the strongest columns will be between the equator and 30S. The equatorial column corresponds to convection, and the 30N column corresponds to the point on the Earth at which the warm air has lost sufficient heat and sinks back to the surface. If 30N is the place where warm air becomes cold again, then this is where heat transport stops with the Hadley cell, and as a result, there should be a strong temperature gradient. From our polar front project, we know that horizontal temperature gradients produce a vertical wind shear. Therefore, the place where we observe the strongest temperature gradient should be where we see the fastest increasing winds and the greatest overall wind speed. Looking at the following plot of zonal (i.e. west-east) winds, the greatest wind shear and resulting wind speed occurs at 30N, where the greatest horizontal temperature gradient is and where the air in the Hadley cell cools off again.
Superimposing the three graphs of wind speed on top of each other, we can get a general picture of how winds move in the Hadley cell between the equator and 30N. Air rises at the equator due to convection, is transported northward to compensate for the meridional heat imbalance, cools off at 30N where the greatest temperature gradient is, resulting in zonal winds and also the sinking of now dense air and resulting equatorward movement to compensate again for the meridional heat imbalance.
We have now shown that what we expected, an overturning cell at low latitudes to transport heat to the mid-latitudes does in fact exist and we can see it by inspecting wind velocity vs. latitude in the atmosphere. The Hadley cell is only responsible for heat transfer to the mid-latitudes, so there must be some other mechanism to bring heat from the mid-latitudes to the pole. The farther away from the equator we go, the more important the Coriolis parameter is, and small-scale eddies or rotating weather systems will be the main mechanism for heat transport. To understand what this looks like, we will conduct our tank experiment again except at a faster rotation rate to make a link to the higher latitudes.
Eddy Heat Transport in the Tank
In the tank experiment mentioned above, we looked at how water moves between the outskirts of the tank and the center where the ice bucket is to transfer heat. A rotation rate was chosen to create movement most similar to the Hadley cell, but we conducted the experiment again except at a faster rotation rate to create movement most similar to eddies. Likewise, four thermistors were placed in the tank when the rotational speed was increased and thus the Coriolis parameter augmented to a point when the regime would shift to one of eddy heat transport. Figure 2.0 illustrates the positions of said thermistors:
Figure 2.0:
It would be anticipated that the greatest temperatures would be found at thermistor 1, and the coolest in turn at thermistor 3 or 4. Figure 2.1 depicts the trace of thermistors' reports during the duration of the experiment:
Figure 2.1:
Though difficult to discern due to the technical limitations of Wiki, it becomes clear that the warmest temperatures were, in fact, recorded at thermistor 1, with the chilliest readings located at thermistor 3. This is exactly what would be expected. Unlike the gentler slope within the laminar fluid, however, the eddy transport nature of this second trial of the experiment lead to an oscillating, varying "stair step" pattern in temperature measurements. In addition, an effort was made to determine the velocity of different points using the particle tracking software; the positions of the points sampled is shown in figure 2.2:
Figure 2.2:
The most dramatic eddy sampled using the particle-tracking software, visible in part on bottom left-hand side of figure 2.2, is quite telling apropos to the "heat flux" within the tank. Despite rotation throughout the eddies towards and away from the center, the thermal gradient within the tank suggests that heat should have a net movement towards the center. A side view of the same graph focused on the most intense eddy illustrates the "winds" embedded within this rotating swirl of fluid; positive velocities are towards the center, with negative velocities away from the center. At first a chaotic scatter of points demarcating velocities at given points (with the x and y axes marking position, with the z axis reserved for velocity), a blue parallelogram has been drawn into Figure 2.3, along with a yellow line illustrating the zero level of velocity from the graph's perspective. One can clearly note that the area of the parallelogram containing data points above the zero line is far greater than below the line, indicating a net movement towards, and thus a heat flux in the direction of, the center. This is exactly what is expected in the atmosphere, and what our group observed in the tank.
Figure 2.3:
In addition, our group was able to note at least half a dozen eddies embedded within our fluid visually using food coloring and dye; the red was placed on the outside of the tank, with blue inside the tank. As mixing occurred, the dyes were dragged along with the rotating fluid, and as such the patterns of warm and cold water were able to be shown. Figure 2.4 depicts these illustrative markings:
Figure 2.4:
Thinking back to Project 2, an investigation into the thermal wind balance and formation of the jet stream, one could in essence convert the motion of particles throughout the fluid into a vector field; in doing so, there would exist one continuous path that completely encircles the central cold-dome within the tank. This would indicate the jet stream, with the serpentine pattern waving towards and away from the center and ensnared within eddies and vortices rotating around in the larger body of fluid. One may note that the blue and red eddies rotate in opposite directions, much as is the case within the atmosphere due to regimes of higher and lower pressure.
One item of note in figure 2.1 is that the temperature at each sensor, though subject to slight oscillations due to eddies, slowly decrease with time somewhat uniformly throughout the fluid. This is to be expected, as the thermal energy contained within the relatively warmer is transferred and over time reduced as it in turn warms the ice to above melting point. Thus, though the temperature of the ice/water solution remains at 32 degrees, the increased thermal energy is instead utilized in the form of latent heat, responsible for the change in phase of the liquid. As such, the amount of energy needed to fully melt the block of ice placed in the center should be equal to product of the ice's mass and the specific heat of fusion. Therefore, considering that 771.4 grams of ice were used in the experiment, one would anticipate that roughly 258,000 Joules of energy would be necessary complete this thermal transaction. Spread over a period of approximately 4,000 seconds, the result is in effect the energy required to power a 64-Watt light bulb, and may be practically thought of as a "negative light bulb" placed in the center of the tank according to Dr. John Marshall. An example of a 65-Watt bulb is depicted in figure 2.5.
Figure 2.5:
Eddy Heat Transport in the Atmosphere
The tank experiment showed us that small-scale rotations are responsible for transporting heat when the rotation of the Earth is important enough as is the case at sufficiently high rotation rates and high latitudes. We will now look for evidence that heat is transported via these eddies or transients as they are called since the systems last for a finite amount of time. Below is a map of the surface of the Earth plotted with transient heat flux. Yellow contours mean northward heat flux, and colder blues mean southward heat flux.
To get a clearer sense of the transient heat flux dependence on latitude, we will take a zonal average i.e. average over all of the longitudes.
With this graph, the general trend of transient heat flux being directed poleward is evident, and the total transient heat flux can be quantized by summing over all pressure levels in the next figure.
Here is a quantitative graph of transient heat flux at different latitudes. Notice that the zonally averaged heat flux map shows that the maximum transient heat flux is ~6 PW, and this graph only shows 1-1.5 PW. A possible reason for this is the fact that the data set filtered out weather systems that were shorter than two weeks. We can also look at the transient heat flux for each pressure level, which shows that the most heat is transferred just above the surface at 850 mb.
Using NCDC's and NCEP's data, the Transient Energy Flux across different levels of the atmosphere in January can be plotted with relative ease. As previously mentioned, the Coriolis parameter of the earth only supports the Hadley cell between 0 and 30º on either side of the equator. As such, the remainder of the globe is dominated by eddy heat transport.
Because January features winter in the northern hemisphere, a stronger temperature gradient leads to a more significant poleward heat transport than in the southern hemisphere. This is evident in our findings, given that the peak at northern latitudes in flux is far greater than in southern latitudes. The Hadley cell theory is also supported by the weak positive flux at the surface (1000 millibars) between 0º and 30º north. This is because there is a weak easterly with a slightly northerly component as air returns radially towards the equator; theoretically, there could be negative flux, but some "mixing" of northerly winds likely aids in some weak northerly flow near the surface. The majority of the heat transport occurs aloft, but the heat flux which measures the amount of heat transport at various levels, should be directly proportional to height (for obvious reasons described previously under the theory of Hadley cells) and pressure (as denser air can carry more heat per unit area). Therefore, the location of the "maximum" should be the "happy medium" between height and pressure, which we would estimate to be located around 700-850 millibars, as revealed by the graph. This is coincident with the jet stream, which is essentially the a narrow band of intense westerlies. In addition, the rising air characteristic of the Intertropical Convergence Zone at the equator is tempered by a weak high pressure dome aloft, which can be seen in the extremely weak flux towards the equator at the uppermost levels of the atmosphere. In addition, it was previously mentioned that friction with the surface would reduce the flux close to the ground, which is supported by the lower level of flux at the 1000 millibar level.
Figure 1.92:
(Please do note that the Transient Heat Flux in Petawatts should not feature a 10^-21 factor; that should be omitted).
Figure 2.4:
Thinking back to Project 2, an investigation into the thermal wind balance and formation of the jet stream, one could in essence convert the motion of particles throughout the fluid into a vector field; in doing so, there would exist one continuous path that completely encircles the central cold-dome within the tank. This would indicate the jet stream, with the serpentine pattern waving towards and away from the center and ensnared within eddies and vortices rotating around in the larger body of fluid. One may note that the blue and red eddies rotate in opposite directions, much as is the case within the atmosphere due to regimes of higher and lower pressure.
One item of note in figure 2.1 is that the temperature at each sensor, though subject to slight oscillations due to eddies, slowly decrease with time somewhat uniformly throughout the fluid. This is to be expected, as the thermal energy contained within the relatively warmer is transferred and over time reduced as it in turn warms the ice to above melting point. Thus, though the temperature of the ice/water solution remains at 32 degrees, the increased thermal energy is instead utilized in the form of latent heat, responsible for the change in phase of the liquid. As such, the amount of energy needed to fully melt the block of ice placed in the center should be equal to product of the ice's mass and the specific heat of fusion. Therefore, considering that 771.4 grams of ice were used in the experiment, one would anticipate that roughly 258,000 Joules of energy would be necessary complete this thermal transaction. Spread over a period of approximately 4,000 seconds, the result is in effect the energy required to power a 64-Watt light bulb, and may be practically thought of as a "negative light bulb" placed in the center of the tank according to Dr. John Marshall. An example of a 65-Watt bulb is depicted in figure 2.5.
Figure 2.5:
Eddy Heat Transport in the Atmosphere
The above graphs shows a vertically averaged plot of transient heat flux on Earth. Yellow/blue corresponds to north/southward flow. Overall, there is a pattern of poleward transient heat blux, and everywhere else, it's zero.
If we average over longitude, we can get a purely latitude dependent graph, which shows how the mid-latitudes are the only regions of transient heat flux.